In recent years, the hyperspectral target detection technique has received widespread attention, and various methods have been proposed. However, most of these methods suffer mainly from two problems. First, the methods perform detection based on the original image, and thus, the targets are easily contaminated by the complex background. Second, a large amount of prior information is usually required, but difficult to obtain. To solve these problems, we proposed a Sparse Tensor model-based Spectral Angle (STSA) detector for hyperspectral target detection. First, with a tensor decomposition model, we obtained a sparse tensor component separated from the original image. Based on the sparse tensor, the detection is less contaminated by the background, and the sparse tensor retains the spatial structure using the developed 3D tensor-based decomposition model. Second, to further enhance the performance, the projected spectral angle method was developed, which only requires a single target spectrum as prior information. Finally, to increase the separation ability between the target and background, we applied a statistical strategy to highlight the target and suppress the background. The experiments were carried out based on four public hyperspectral datasets. The results showed that the proposed STSA method is more accurate than 11 benchmark methods.

We consider a sequence of four variable polynomials by refining Stieltjes's continued fraction for Eulerian polynomials. Using combinatorial theory of Jacobi-type continued fractions and bijections, we derive various combinatorial interpretations in terms of permutation statistics for these polynomials, which include special kinds of descents and excedances in a recent paper of Baril and Kirgizov. As a byproduct, we derive several equidistribution results for permutation statistics, which enables us to confirm and strengthen a recent conjecture of Vajnovszki and also to obtain several companion permutation statistics for two bistatistics in a conjecture of Baril and Kirgizov.

An ordered partition with k blocks of $[n]:=\{1,2,\ldots, n\}$ is a sequence of k disjoint and nonempty subsets, called blocks, whose union is $[n]$. Clearly the number of such ordered partitions is $k!S(n,k)$, where $S(n,k)$ is the Stirling number of the second kind. A statistic on ordered partitions of $[n]$ with k blocks is called an Euler–Mahonian statistic if its generating polynomial is $[k]_q!S_q(n,k)$, which is a natural q-analogue of $k!S(n,k)$. Motivated by Steingrímsson's conjectures dating back to 1997, we consider two different methods to produce Euler–Mahonian statistics on ordered set partitions: (a) we give a bijection between ordered partitions and weighted Motzkin paths by using a variant of Françon–Viennot's bijection to derive many Euler–Mahonian statistics by expanding the generating function of $[k]_q!S_q(n,k)$ as an explicit continued fraction; (b) we encode ordered partitions by walks in some digraphs and then derive new Euler–Mahonian statistics by computing their generating functions using the transfer-matrix method. In particular, we prove several conjectures of Steingrímsson.

Two formulas of combinatorial nature already derived by means of analytical methods are proved combinatorially. They provide the calculation of the linearization coefficients of products of orthogonal polynomials (Hermite, Laguerre, Charlier, Krawtchouk, Meixner). On the other hand, one of them specializes into the two formulas of Pfaff–Saalschütz and Dixon.